I drive past this building every day on my way to work. It is Young Tower at 6080 Young Street in Halifax. I think it is pretty interesting... I used this picture as a problem solving warm up activity for a group of grade 10 math teachers recently. I gave each group of teachers a large piece of chart paper and asked them to divide the paper in half with a line. I asked teachers to brainstorm what they notice about this picture and record it on one half of their chart paper. I asked them to look at the picture using a number of lenses. What would an architect notice about this image? What would a person who worked at this building notice about this picture? What would a mathematician notice about this picture? After about 5 minutes of brainstorming, I asked each group to tell me one thing they noticed and I recorded it at the front of the room. Groups noticed things like the number and size of windows on the building ("about half the lateral surface is glass"), the shape of the building ("almost a cube"), the picture must have been taken on a weekend because there are very few cars in the parking lot, and the weather was really nice that day. Next I asked them to brainstorm what they wonder about this picture and record in on the other half of their chart paper. If this picture was the start of a math problem, what could that math problem be? What things that they noticed sparked their curiosity? After another 5 minutes, I asked each group once again to tell me one thing that they wondered. After looking at all the questions that the groups posed, we selected one and asked everyone to estimate an answer to that question. I also asked them what information would they need to make a more accurate estimate. Once they had an initial estimate, I gave them some additional information about the building and let them revise their estimate. We had several really interesting questions posed by groups. Some questions concerned the shape of the building, like "How close to a perfect cube is this building?" Other questions focused on finance such as, "How much revenue is generated by leasing all of the office space in this building?" One of my favourite 'wonderings' was, "How much wrapping paper would it take to wrap this building up like a Christmas present?"

​​The building is a square based prism with each side measuring 120’8”. The height of this building is 128 ft.

Each plate of glass is 5’ wide and approximately 5’ tall.

Building has 10 floors.

Each floor has approx. 14,400 square ft.

The building has a total of 140,793 sq ft of leaseable area.

This "I Notice/I Wonder" problem solving strategy is one that I saw shared by Max Ray-Riek from the Math Forum. He has a blog where he talks about Noticing and Wondering in High School. This strategy starts off with brainstorming to let students get familiar and engaged with a problem situation before jumping into a specific question to solve. By having students come up with questions, you'll often get more engagement and interest. It also allows you to respond to interesting suggestions from students that you might not have considered. It allows everyone in the class meaningful participation in the conversation because everyone has something that they can notice. This strategy might also create additional opportunities for differentiation by using several different questions that students suggested.

One of my favourite activities recently has been Fawn Nguyen's Snap Hotel. This activity can also be found on the NCTM Illuminations website. You might see this activity online under a few different names... my favourite is Hotel^3. It is an engaging problem solving task. Students are given 50 multi-link cubes and instructed to build a model hotel. Each cube represents one hotel room. Some rooms are more desirable than others and can be rented for higher prices. The number of windows and whether or not the room has a roof determine the rental price. Students also have to consider costs associated with property and height. The hotels that students create in this activity remind me of Montreal's Habitat 67. Students might talk about the costs and benefits of this type of architecture. This activity can be used to assess a number of Nova Scotia mathematics curriculum outcomes.

The Rules - As a team, build a hotel that yields the highest profit.

Each cube represents a hotel room.

All 50 cubes must be used.

Entire hotel is one piece.

Hotel must stand freely on at least one side of cube.

All rooms must have at least one window, a window is any exposed vertical side of cube.

Below are some hotel's created by teachers during a session at the NS MTA conference in Oct. 2015.

Hotel Expenses•Land costs $400 per square unit.•Each roof costs $10 each, roof is any exposed top side of cube.•Each window costs $5 each, a window is any exposed vertical side of cube.•Height charge is #floors x $300

One suggestion to improve this task is to overhaul the height tax in the original version. Students see the jump from a 10 floor building (50% of land cost) to an 11 floor building (1000% of land cost) and they immediately conclude that a tall structure is out of the question. They don't even explore the option of a tall, 50 floor tower, despite it being quite simple and quick to test it's profit. Instead of the height tax being a percentage of the land cost, I would make it a flat rate based on the number of floors... for example, height cost = # floors * $300. Now instead of a huge jump from 10 floors to 11 floors, there is a gradual increase as a building grows in height.

Another suggestion might be to include on the rubric a score for aesthetic appeal. A unique, interesting, or environmentally friendly building design might induce more residents to stay at the hotel and thus increase its revenue. After all, you can't make money with a hotel unless people want to stay there.

Resources​Below are my Powerpoint introduction and a student handout. Also included is an analysis of several different hotels to see how their profits compare.

This task is easy to modify or differentiate... you could change the number of cubes (as few as 10-15 cubes for younger kids, and maybe up to 100 cubes for high school students)... you could increase/reduce the number of constraints or change any of the costs/expenses.

Students work in small groups and have the opportunity to test out a number of different structures. They can continue to create, test, and refine their model.

This activity is easy to set up and put away and doesn't require any materials besides fairly common multi-link cubes.

Students can take the role of an architecture firm and explain why their design should be chosen... perhaps they will focus on the profit of their structure, the aesthetics, or the livability. They could watch the TEDx talk from Moshe Safdie, the architect of Habitat 67.

Update: If I did the activity again in a classroom, I don't think I could resist having a monster made of multi-link cubes (from Gigo Blocks) scaling the side of the building like a giant Godzilla! Lots of fun.

Nova Scotia Curriculum Outcomes

MT8 - G01Studentswill be expected to draw and interpret top, front, and side views of 3-D objects composed of right rectangular prisms.

MT9 – G01 Students will be expected to determine the surface area of composite 3-D objects to solve problems.

MT10 - FM01 and MTW10 – N01 Students will be expected to solve problems that involve unit pricing and currency exchange, using proportional reasoning.

MTW10 - G01 Students will be expected to analyze puzzles and games that involve spatial reasoning, using problem-solving strategies.

MTW11 - G03Students will be expected to model and draw 3-D objects and their views.

MTW12 - N03 Students will be expected to critique the viability of small business options by considering expenses, sales, and profit or loss.